Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. cos² 105° - sin² 105°

검증된 단계별 안내

Recognize that the expression \( \cos^2 105^\circ - \sin^2 105^\circ \) matches the form of the cosine double-angle identity: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).

Rewrite the given expression using the double-angle identity: \( \cos^2 105^\circ - \sin^2 105^\circ = \cos(2 \times 105^\circ) \).

Calculate the angle inside the cosine function: \( 2 \times 105^\circ = 210^\circ \), so the expression becomes \( \cos 210^\circ \).

Recall the exact value of \( \cos 210^\circ \) by considering its reference angle and quadrant. The reference angle is \( 210^\circ - 180^\circ = 30^\circ \), and since 210° is in the third quadrant where cosine is negative, \( \cos 210^\circ = -\cos 30^\circ \).

Use the exact value \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) to write \( \cos 210^\circ = -\frac{\sqrt{3}}{2} \).

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Double-Angle Identities

Double-angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For cosine, the identity cos(2θ) = cos²θ - sin²θ directly relates to the given expression, allowing simplification by recognizing the pattern.

Exact Values of Trigonometric Functions

Exact values refer to precise trigonometric values for special angles, often expressed in terms of square roots and fractions. Knowing or deriving exact values for angles like 105° (which can be expressed as 60° + 45°) is essential for finding the exact value of the expression.

Angle Sum and Difference Formulas

These formulas allow the calculation of sine, cosine, or tangent of sums or differences of angles. For example, cos(105°) can be found using cos(60° + 45°) = cos60°cos45° - sin60°sin45°, which helps in determining exact trigonometric values needed for the problem.